6.7.1 Tensor Type Ordered Pair

A Tensor Type Ordered Pair structures tensor components with type-specific order, defining algebraic relationships in multilinear algebra.

Tensor Type Ordered Pair is the formal identification of a tensor's type with an element (p, q) of the set N × N of ordered pairs of non-negative integers, taken as the primary, defining object of type classification rather than as a derived summary of some more detailed structure; two tensors are assigned the same type precisely when their associated pairs are equal as elements of N × N, (p, q) = (p', q') if and only if p = p' and q = q'. This formalization is what allows type to be treated as a genuine label — usable for indexing, comparison, and organizing families of tensor spaces — rather than as an informal description.

Type as an Element of N × N

The Defining Set

Type classification assigns to every tensor a pair belonging to the set of ordered pairs of non-negative integers:

type (T) \in N \times N = {(p, q) : p, q \in {0, 1, 2, \dots}}

with p and q respectively the contravariant and covariant slot counts.

Equality of Types as Equality of Pairs

Two tensors, possibly of entirely different orders classification would otherwise have to describe separately for each component, are said to have the same type exactly when their pairs agree coordinate by coordinate:

(p, q) = (p^{'}, q^{'}) \Leftrightarrow p = p^{'} and q = q^{'}

reducing the question "do these two tensors share a type" to the elementary question of whether two ordered pairs of integers are equal.

Why a Pair, Rather Than a Single Number or a Set, Is the Right Encoding

A single combined number, such as the order p + q, would conflate different types sharing the same total; an unordered set {p, q} would conflate (p, q) with (q, p) even when p ≠ q; the ordered pair is the minimal encoding that avoids both forms of conflation while adding no extraneous information beyond the two counts themselves.

The Pair as an Indexing Coordinate

Parameterizing Families of Tensor Spaces

The ordered pair is used directly as an index for the space of all type (p, q) tensors over a vector space V, commonly written T^p_q(V), so that varying p and q sweeps through an entire family of related spaces, one for each point of N × N:

T_{q}^{p} (V) = \underset{p}{\otimes} V^{*} \otimes \underset{q}{\otimes} V

Diagram of the Type Lattice as an Indexing Grid

Comparing Types by Comparing Pairs Coordinatewise

The ordered pair also supports a natural partial order, (p, q) ≤ (p', q') when p ≤ p' and q ≤ q', which is used to compare the relative "size" of two types, for example to note that T^1_1(V) sits strictly below T^2_2(V) in the sense that every operation available on the smaller type extends to an analogous operation on the larger one obtained by tensoring with additional identity-like factors.

Alternative Encodings and Why They Are Set Aside

Order Plus a Single Bit Is Insufficient Beyond p, q ≤ 1

One might attempt to encode type using the order k = p + q together with a single bit indicating "more upper than lower," but this fails to distinguish, for instance, type (3, 1) from type (2, 2), both of order 4 with more upper than lower slots (for the latter, tied) inadequately captured by one bit; the full ordered pair is needed once p and q can each independently exceed 1.

The Difference p − q Loses Information

Encoding type by the single integer p - q also fails, since (3, 1) and (2, 0) both give a difference of 2 despite being genuinely different types with different total slot counts; the ordered pair retains both totals simultaneously, which no single derived number can do without loss.

Why Treating Type as This Specific Object Matters

Enabling Rigorous Comparison and Indexing

Fixing type as an element of N × N rather than as an informally described property makes statements like "these two tensors have the same type" and "this space is indexed by type (2,1)" precise enough to support formal proof and systematic organization of tensor spaces, rather than remaining at the level of descriptive convenience.

A Foundation Consistent With Valence's Refinement of Type

Defining type formally as this ordered pair is exactly what makes it meaningful to later say that valence classification refines type: valence adds positional information on top of a type that is already well-defined as a specific pair, with no ambiguity about what, precisely, is being refined.